Modal parameter estimation using free response measured by a continuously scanning laser Doppler vibrometer system with application to structural damage identification

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Abstract

Spatially dense vibration measurement can be obtained by use of a continuously scanning laser Doppler vibrometer (CSLDV) system that sweeps its laser spot along a scan path. For a linear, time-invariant, viscously damped structure undergoing free vibration, a type of vibration shapes called free response shapes was defined and obtained by the authors using a CSLDV system with the demodulation method. To date, application of free response shapes is limited to structural damage identification, and they cannot be directly used for model validation while mode shapes can be. This paper extends the concept of free response shapes by proposing a new modal parameter estimation (MPE) method using a CSLDV system to estimate modal parameters of the structure undergoing free vibration, including natural frequencies, modal damping ratios, and mode shapes; the MPE method is applicable to linear structures without repeated and closely spaced modes. Advantages of the proposed method are: (1) modal damping ratios and mode shapes can be accurately estimated from obtained free response shapes in the least-square sense, (2) the scanning frequency of the CSLDV system can be relatively low, and (3) estimated mode shapes can be used for structural damage identification as if they were measured by stepped scanning of a scanning laser Doppler vibrometer. A baseline-free method is applied to identify structural damage using mode shapes estimated by the proposed MPE method. The method does not require any baseline information of an undamaged structure, such as its complete geometry, material properties, boundary conditions, modal parameters, and operating deflection shapes. In the proposed MPE method, natural frequencies of the structure are identified from free response of certain fixed points on the structure; its modal damping ratios and mode shapes are simultaneously estimated using free response shapes measured by a CSLDV system. Both numerical and experimental investigations are conducted to study the MPE method and its application to baseline-free damage identification with mode shapes estimated by the MPE method.

Introduction

Vibration-based damage identification has been a major research topic of structural dynamics in the past few decades [1,2]. Occurrence of damage in a structure undermines its capability of supporting design loads and can result in its excessive deformation, which is attributed to changes in its structural properties, such as its stiffness. One assumption of a vibration-based damage identification method is that occurrence of damage changes modal parameters of a structure, including natural frequencies, modal damping ratios, and mode shapes, which can be accurately estimated by modal analysis [3]. Accurately estimated modal parameters can also assist model validation and updating.

A continuously scanning laser Doppler vibrometer (CSLDV) system is an ideal instrument for modal parameter estimation (MPE) as it is capable of accurate, non-contact and temporally dense vibration measurement and also capable of spatially dense mode shape measurement [4]. A CSLDV system consists of three key components: a laser Doppler vibrometer, a scanner and a controller [5]. The vibrometer measures the velocity of a point on a test structure where its laser spot is located. The laser beam of the vibrometer is directly shined onto first-surface mirrors of the scanner and the spot is continuously swept along a prescribed scan path on the structure by rotating the mirrors that are controlled by the controller. While the spot is continuously swept, velocity at each measurement point on the scan path is measured at the instant when the spot arrives at the measurement point, and the number of the measurement points can be tens and even hundreds of thousands, depending on the sampling and scan frequencies of the vibrometer. A CSLDV system has been successfully used for modal analysis and measurements of vibration shapes, such as mode shapes [[6], [7], [8]] and operating deflection shapes [[9], [10], [11], [12], [13]], which can be achieved with high accuracy.

A CSLDV system has been used to measure high-fidelity vibration shapes of structures undergoing steady-state vibrations for damage identification [5,14], and the vibration shapes measured by a CSLDV system can be used to identify structural damage as small as notch-size ones [15]. A type of vibration shapes called free response shapes was defined and measured by a CSLDV system when a linear underdamped beam underwent free vibration [16]. Free response shapes were defined to identify structural damage, where damage indices associated with multiple elastic modes of a beam could be obtained. A free response shape is different from a mode shape, since the former is time-varying with decaying amplitudes and the latter is not. So far, application of free response shapes is limited to structural damage identification and they cannot be directly used for model validation and updating due to two reasons. One is that a free response shape has an amplitude that is determined by excitation. Unless one can accurately measure the excitation, a free response shape cannot be used for model validation and updating. Another reason is that modal damping ratios cannot be estimated from free response shapes that are obtained in the method in Ref. [16]. An experimental modal analysis method was proposed [8], where excitation to a test structure and its free response measured by a CSLDV system yielded pseudo-frequency response functions of the structure, which were used to estimate modal parameters of the structure. In this method, the measured response is lifted to each measurement point as if the response were measured in a pointwise manner. A limitation of the method is that measured mode shapes of modes with relatively high natural frequencies can have low qualities due to speckle noise caused by a relatively high scanning frequency, which is needed since the scanning frequency of the CSLDV system is equal to the sampling frequency of the lifted response at each measurement point.

In this work, derivation of free response shapes of a linear, time-invariant, viscously damped structure undergoing free vibration is shown. A new MPE method using free response measured by a CSLDV system is proposed to accurately estimate modal parameters of the structure with a step-by-step procedure. The proposed MPE method is novel as it extends the concept of free response shapes of the structure to simultaneously estimate its modal damping ratios and mode shapes and the estimated modal damping ratios and mode shapes can be used for model validation and updating and structural damage detection. A baseline-free non-model-based damage identification method is applied to identify structural damage in a structure. The method does not require any baseline information of an undamaged structure, such as its complete geometry, material properties, boundary conditions, modal parameters, and operating deflection shapes. In the damage identification method, a curvature damage index (CDI) is obtained by comparing a curvature mode shape, which corresponds to a mode shape estimated by the MPE method, with that from a polynomial; the polynomial fits the mode shape estimated by the MPE method. Structural damage can be identified in neighborhoods with consistently large CDIs corresponding to multiple modes. A numerical investigation is conducted to study the MPE method and application of the damage identification method. An experimental investigation was also conducted to validate the MPE method and application of the damage identification method by using data from a test for free response shapes in Ref. [16].

The remaining part of this paper is outlined as follows. Derivation of free response shapes is presented in Secs. 2.1 Free response of a damped structure, 2.2 Free response shapes, the new MPE method using a CSLDV system is proposed in Sec. 2.3, and the structural damage identification method is presented in Sec. 2.4. Numerical and experimental investigations of the MPE method and baseline-free method are presented in Secs. 3 Numerical investigation, 4 Experimental investigation, respectively. Finally, conclusions of this study are presented in Sec. 5.

Section snippets

Free response of a damped structure

Free response in the form of the displacement of a linear, time-invariant, viscously damped structure can be obtained by solving its governing partial differential equation:B2zx,t2t+Czx,tt+Lzx,t=0,xD,t0where B·, C· and L· are a mass operator, a damping operator and a stiffness operator, respectively, z is the displacement of the structure at the spatial position x at time t, and D is its spatial domain. Boundary and initial conditions of the structure are known. Note that the initial

Numerical investigation

A finite element model of a damaged aluminum cantilever beam with a length L = 0.8 m, Young's modulus of 68.9 GPa, a mass density of 2700 kg/m3 and a damping coefficient of Kelvin-Voigt damping model of 8 × 10−7 s is constructed using ABAQUS. The beam has a uniform square cross-section with a side length of 0.01 m. The damage is in the form of thickness reduction, which is located between x=616L and x=716L, where x is the position of a point on the beam. The damaged portion of the beam has a

Experimental investigation

An experimental investigation was performed in Ref. [16] to obtain free response shapes of a damaged cantilever beam and identify its damage. Experimental data and results from the investigation, including free response measurements of the beam under different excitation methods, which were measured by a CSLDV system with different scan frequencies, natural frequencies and free response shapes of the beam, were used here to perform the proposed MPE method and identify the damage. In this

Conclusions

Derivation of free response shapes of a linear, time-invariant, viscously damped structure is shown. The only current application of free response shapes is structural damage identification and they cannot be used for model validation and updating. A new MPE method using free response measured by a CSLDV system is proposed to estimate modal parameters, including natural frequencies, modal damping ratios, and mode shapes based on the concept of free response shapes. A critical assumption of the

CRediT authorship contribution statement

Y.F. Xu: Methodology, Funding acquisition, Software, Writing - original draft. Da-Ming Chen: Software, Validation. W.D. Zhu: Conceptualization, Methodology, Funding acquisition, Writing - review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

The authors are grateful for the financial support from the National Science Foundation through Grant Nos. CMMI-1335024, CMMI-1763024, CMMI-1762917 and the College of Engineering and Information Technology at the University of Maryland, Baltimore County through a Strategic Plan Implementation Grant. The first author is also grateful for the faculty startup support from the Department of Mechanical and Materials Engineering at the University of Cincinnati.

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